Page 1

Nanoscale heat flux between nanoporous

materials

S.-A. Biehs

Laboratoire Charles Fabry, Institut d’Optique, CNRS, Universit´ e Paris-Sud, Campus

Polytechnique, RD128, 91127 Palaiseau Cedex, France

P. Ben-Abdallah

Laboratoire Charles Fabry, Institut d’Optique, CNRS, Universit´ e Paris-Sud, Campus

Polytechnique, RD128, 91127 Palaiseau Cedex, France

F. S. S. Rosa

Laboratoire Charles Fabry, Institut d’Optique, CNRS, Universit´ e Paris-Sud, Campus

Polytechnique, RD128, 91127 Palaiseau Cedex, France

K. Joulain

Institut P’, CNRS-Universit´ e de Poitiers UPR 3346, 86022 Poitiers Cedex, France

J.-J. Greffet

Laboratoire Charles Fabry, Institut d’Optique, CNRS, Universit´ e Paris-Sud, Campus

Polytechnique, RD128, 91127 Palaiseau Cedex, France

Abstract:

Garnett description for effective media we study the radiative heat transfer

between two nanoporous materials. We show that the heat flux can be sig-

nificantly enhanced by air inclusions, which we explain by:(a) the presence

of additional surface waves that give rise to supplementary channels for heat

transfer throughout the gap, (b) an increase in the contribution given by the

ordinary surface waves at resonance, (c) and the appearance of frustrated

modes over a broad spectral range. We generalize the known expression for

the nanoscale heat flux for anisotropic metamaterials.

By combining stochastic electrodynamics and the Maxwell-

© 2011 Optical Society of America

OCIS codes: (160.1190) Anisotropic optical materials; (240.5420) Polaritons

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Page 3

1.Introduction

Near field heat transfer [1, 2, 3, 4] between closely spaced isotropic media has been inten-

sively studied since it has been predicted that the heat flux at nanoscale can exceed the far-

field limit of the Planck’s blackbody theory by orders of magnitude [5, 6]. When consid-

ering dielectrics, surface phonon polaritons provide additional enhancement as discussed in

Refs. [7, 8, 9]. Several experiments have recently confirmed the theoretical predictions for sim-

ple systems [15, 16, 11, 13, 14].

With the modern techniques of nanofabrication it is now possible to explore a whole new

level of complexity in material science and to fabricate artificial materials that can exhibit a

considerable diversity of optical properties [17, 18, 19, 20, 21, 22]. In many situations, these

composite media possess privileged orientations so that their electromagnetic response depends

on the direction of photons propagation. When the photon’s wavelength in such a medium is

large compared to the size of its representative unit cell, the latter behaves effectively like an

anisotropic material and therefore may be described by an effective permittivity tensor (and,

when necessary, an effective permeability as well). This naturally points to the question of how

anisotropy influences the near-field heat transfer.

In this work, we address this question in the particular case of two semi-infinite uniaxial

media characterized by optical axes orthogonally oriented with respect to the surface of inter-

action. The paper is organized as follows: In Sec. II we derive the expression for the heat flux

between two anisotropic media. After a brief description of the relevant composite media to

our purposes in Sec. III we investigate in Sec. IV the surface and Brewster modes supported

by them and their main features. Next, we compare in Sec. V the near-field heat exchanges

between two uniaxial media to the classical ones between two isotropic media. Finally, in order

to explain the difference in the behavior of isotropic and anisotropic materials, we discuss in

Sec. VI the transmission factor in detail between two uniaxial media and in Sec. VII we present

our conclusions.

2.Radiative heat transfer between anisotropic media

Let B1and B2be two anisotropic semi-infinite bodies, filling respectively the regions z < 0

and z > d and leaving a vacuum gap between them. In order to ensure a stationary process, we

assume that the Biare in local thermal equilibrium at a temperature Ti, with T1?= T2. The heat

flux P between the two bodies is given by

P(T1,T2,a) =

?

A12

dA·?S?,

(1)

where S = E×H is the Poynting vector and A12is any surface that separates the two bodies.

By taking such a surface to be a plane defined by z = z0(0 < z0< d) and using the (transverse)

translational invariance of our system, the previous equation simplifies to

P(T1,T2,a) = A?Sz?,

(2)

showing that only the z-component of the Poynting vector is needed. After a straightforward

calculation, the latter can be conveniently written as [3]

?Sz? =

?∞

0

dω

2π

?Θ(ω,T1)−Θ(ω,T2)??Sω?,

(3)

where we identify the mean energy of a harmonic oscillator

Θ(ω,T) =

¯ hω

¯ hω

kBT−1

e

,

(4)

Page 4

and also the averaged spectral Poynting vector [3]

?Sω? = 2ReTr

??

Adr′

?

?

G(r,r′)∂z∂′

zG†(r,r′)−∂zG†(r,r′)∂′

zG(r,r′)

??

z′=z=z0

.

(5)

where r = r?+zˆ z and G(r,r′) is the electrical Green’s dyadic, satisfying

∇×−ω2

?− →

∇×− →

c2ε(r,ω)

?

G(r,r′,ω) = δ(r−r′)I.

(6)

Moreover, we have introduced Boltzmann’s constant kB, Planck’s constant 2π¯ h; the † symbol-

izes hermitian conjugation and Tr the 3×3 trace.

In order to evaluate the heat flux in the given geometry we have to determine the Green’s

dyadic inside the gap region. This can be done by considering the multiple scattering of a plane

wave due to a source inside the gap [23]. Details and the final expression for the Green’s dyadic

can be found in appendix A. When inserting the resulting expression in Eq. (40) into the heat

flux formula, we find

?

The integral is carried out over all transverse wave vectors κ = (kx,ky)tincluding propagating

modes with κ < ω/c and evanescent modes with κ > ω/c, where c is the velocity of light in

vacuum. The energy transmission coefficient T(ω,κ;d) is different for propagating and evanes-

cent modes and can be stated as

?Sω? =

d2κ

(2π)2T(ω,κ;d).

(7)

T(ω,κ;d) =

?

Tr?(1−R†

2R2)D12(1−R1R†

2−R2)D12(R1−R†

1)D12†?,

κ < ω/c

κ > ω/c

Tr?(R†

1)D12†?e−2|γr|d,

(8)

where γr=?ω2/c2−κ2and R1, R2are the 2×2 reflection matrices characterizing interfaces.

?

we see that their elements rλ,λ′

i

are the reflection coefficients for the scattering of an incoming

λ-polarized plane wave into an outgoing λ′-polarized wave. Note that expression (8) is very

general, as it in principle applies to any crystallographic anisotropy, both electric and magnetic.

In the isotropic limit they reduce to the usual Fresnel coefficients

By writing them a bit more explicitly,

Ri=

rs,s

rp,s

i(ω,κ)

i(ω,κ)

rs,p

rp,p

i(ω,κ)

i(ω,κ)

?

,

(9)

rs,s

i(ω,κ) =γr−?εi(ω)ω2/c2−κ2

i(ω,κ) =εi(ω)γr−?εi(ω)ω2/c2−κ2

rs,p

γr+?εi(ω)ω2/c2−κ2,

εi(ω)γr+?εi(ω)ω2/c2−κ2,

rp,p

i(ω,κ) = rp,s

i(ω,κ) = 0,

(10)

and we see that the matrices become diagonal. In addition, we have introduced the matrix D12,

defined by

D12= (1−R1R2e2iγrd)−1,

which gives rise to a Fabry-P´ erot-like denominator for T(ω,κ;d) in the isotropic case.

(11)

Page 5

From Eqs (3), (7) and (8) we see that, once the reflection matrices are known, it is possi-

ble to determine the heat flux between two arbitrary anisotropic semi-infinite bodies kept at

fixed temperatures T1and T2. Moreover, in order to have an independent check, we verified

that Eq. (7) also can be derived from the general scattering formalism derived on Ref. [24].

In the following we will use these expressions to discuss the heat flux between two uniaxial

anisotropic materials with their optical axes normal to the interface.

Figure 1. Sketch of two porous slabs with different temperatures separated by a vacuum

gap.

3.Porous Materials

The structures investigated in this paper are depicted in Fig. 1. They are two semi-infinite media

composed by a host isotropic material, defined by its complex dielectric function εh(ω) =

ε′

orthogonal to the surface as shown in Fig. 1. These inclusions in turn are filled by a medium of

dielectricpermittivityεi,thatisalsoassumedtobeisotropic.Whenthesizeoftherepresentative

unit cell is much smaller than all the other characteristic scales involved, a suitable volume

average of the material’s local electromagnetic response can be made. In our case, the emerging

azimuthal symmetry in this long wavelength limit gives rise to effective uniaxial crystals with

a permittivity tensor of the form

h(ω)+iε′′

h(ω)(whereε′′

h(ω)>0),withuniformcylindricalinclusionsorientedinthedirection

ε = ε?[ex⊗ex+ey⊗ey]+ε⊥ez⊗ez

(12)

where ex, ey, and ezare orthogonal unit vectors in x, y, and z direction. The parallel and per-

pendicular components can be derived from the Maxwell-Garnett effective medium theory

(EMT) [25, 26]

ε?= εhεi(1+ f)+εh(1− f)

ε⊥= εh(1− f)+εif,

εi(1− f)+εh(1+ f),

(13)

(14)

where f is the volume fraction of inclusions. For the structure considered in this work the

deviation from the exact result of homogenization given in Refs. [28, 29] is small even for

relatively high filling factors such as f = 0.5. Hence, we will discuss the heat flux between

porous media with the Maxwell-Garnett expression for f ∈ [0,0.5] in this work.

Page 6

The condition of long wavelengths sets a limit to the lattice constant a of the inclusions for

which the EMT can be used. In the far-field regime this condition is fullfilled when the thermal

wavelength λth= ¯ hc/kBT is much larger than a. In the near-field region the contributing modes

at a distance d above the porous material have a lateral wavelength which depends on d. For

κ = 2π/a (which corresponds to a lateral wavelength a) the evanescent waves are damped as

exp[−?(2π/a)2−ω2/c2d] ≈ exp(−(2π/a)d) in the non-retarded near-field region above the

waves with lateral wavelength larger than a if d > a/(2π). On the other hand, one can argue

that a nonlocal model for the permittivity is necessary if the lateral wave vectors κ are on the

order of π/a. Since the exponential in the transmission coefficient in Eq. (8) for κ > ω/c sets a

cutoff for κ of the modes contributing to the near-field heat flux which is ≈ 1/d, one finds that

a local EMT description is permissible if d ≫ a/π. Hence, for a given lattice constant a of the

inclusions the validity of the EMT in Eq. (13) in the near field regime is restricted to d ≫ a/π.

Artificial structures as depicted in Fig. 1 can have an a on the order of 100nm [22] so that the

distances for which the EMT can be considered as appropriate in this case are about d >30nm.

Nonetheless, chemically produced nanoporous materials can show smaller structures [27] so

that we will consider distances d ∈ [10nm,100µm].

porous material. It follows that the contribution to the heat flux is dominated by evanescent

5

/10

κ

m−1

51015

5

/10

κ

m−1

510 15

5

/10

κ

m−1

(a)

(b)

(c)

5 1015

2

1.8

1.6

1.4

1.2

1.2

1.4

1.6

1.8

2

ω/10

14

ω/10

14

ω/10

14

s−1

s−1

s−1

1.2

1.4

1.6

2

1.8

Figure 2. Plot of the dispersion curves (white dashed lines) from Eq. (22) in the (ω,κ)

plane for filling factors (a) f =0.1, (b) f =0.3, and (c) f =0.5. The white dash-dotted line

represents the light line in vacuum (ω = κc). Furthermore the dark (blue) areas mark the

region for which γpis purely real, whereas the bright (red) areas are the regions for which

γpis purely imaginary.

Page 7

5

/10

κ

m−1

51015

5

/10

κ

m−1

5

/10

κ

m−1

(a)

(b)

(c)

51015

2

1.8

1.6

1.4

1.2

1.2

1.4

1.6

1.8

2

510 15

ω/10

14

ω/10

14

ω/10

14

s−1

s−1

s−1

1.2

1.4

1.6

2

1.8

Figure 3. Plot of ln(1/|rp,p|2) in (ω,κ) plane for (a) f = 0.1, (b) f = 0.3, and (c) f = 0.5.

4. Surface and Brewster modes in porous Media

Let us study the surface waves supported by these media when they are sufficiently far away

from each other so that any coupling of evanescent waves can be neglected. By definition, these

surface waves are resonant surface modes and therefore are determined by the poles of the

reflection coefficients of these media. For out-of-plane uniaxial media the components of the

reflection matrix are

rs,s(ω,κ) =γr−γs

γr+γs,

(15)

rp,p(ω,κ) =ε?γr−γp

ε?γr+γp,

(16)

rs,p= rp,s= 0,

(17)

where γs,pare given by the solutions of Fresnel equations in the anisotropic material [30]

γs=

?

?

ε?ω2/c2−κ2,

ε?ω2/c2−ε?

(18)

γp=

ε⊥κ2,

(19)

Page 8

and hence it follows at once that the surface modes are determined by

(γr+γs) = 0,

(ε?γr+γp) = 0.

(20)

(21)

It is straightforward to verify that in this case only the second equation above can be satisfied,

meaning that only p-polarized surface waves can exist at the interface of these media. Solving

that equation explicitly for κ gives us the sought dispersion relation of surface waves

κ =ω

c

?

ε⊥(ε?−1)

ε?ε⊥−1.

(22)

but one must be aware that (22) has two branches, and only one is connected to surface modes1.

Since their dispersion relation involves ε?and ε⊥, these waves are also called extraordinary

surface waves [31], and they reflect the material anisotropy. When ε?=ε⊥=ε, Eq. (22) degen-

erates into the well-known dispersion relation κ =ω/c?ε/(ε +1) of surface modes supported

Fig. 2 we plot the dispersion curves for silicon carbide (SiC) with vacuum inclusions for differ-

ent filling factors f = 0.1, f = 0.3 and f = 0.5. The dielectric function of SiC is described [32]

by the simple model

εh= ε∞ω2−ω2

ω2−ω2

where ωL= 1.827·1014s−1, ωT= 1.495·1014s−1, Γ = 0.9·1012s−1, and ε∞= 6.7 denote

respectively the longitudinal and transversal optical phonon pulsation, the damping factor and

the high frequency dielectric constant, respectively. In order to avoid the inherent difficulties

of multiple possible interpretations of complex dispersion relations [33], we have deliberately

neglected the host material losses to represent these curves. The relevance of this approximation

can be checked by comparing Fig. 2 with Fig. 3, where we plot the reflection coefficients of

dissipating porous material. In order to distinguish between evanescent and propagative waves

inside the effective medium, solutions of Eq. (22) are superimposed in Fig. 2 to a two-color

background. This background is a binary representation of ζ = sgn(ε?ω2/c2−ε?κ2/ε⊥). In

the blue zones ζ < 0 so that only evanescent modes can exist, and conversely, in the red zones

we have ζ > 0 and all modes are propagative. Similarly the light line ω = cκ allows us to

distinguish between the radiative (propagative) and the non-radiative (evanescent) modes inside

the vacuum. Notice that, in order to satisfy Eq. (22) both ζ and sgn(ω2/c2−κ2) must be the

same. In other words, frustrated modes cannot satisfy the dispersion relation (22).

Now let us turn to the description of modes supported by our artificial structures. For low

filling factors we note in Fig. 2 (a) the existence of two surface modes. The first one (at a lower

frequency) corresponds to the classical surface phonon-polariton (SPP) supported by a massive

SiCsample[34].ThatsurfacemodeisalsopresentinisotropicSiC.Themostinterestingfeature

ofFig.2(a)is,however,theappearanceofasecondsurfacemodeathigherfrequencies,because

it is a signature of the anisotropic character of the material and therefore a direct consequence

of the vacuum inclusions in the host medium. As the porosity increases, both surface waves

split. Beyond a critical filling factor between f = 0.3 and f = 0.5, the upper surface wave

disappears as is seen in Fig. 3. Nevertheless the SPP which still exists continue to move toward

the smaller frequencies, i.e., to ωT. Above the light line, we see that the anisotropy gives rise

to two different types of Brewster modes. At high frequency we recognize the usual modes

by a semi-infinite isotropic medium (bounded by vacuum) with a dielectric permittivity ε. In

L−iωΓ

T−iωΓ

(23)

1The other branch is connected with the so called Brewster modes [33], that are propagating waves for which rp,p

vanishes.

Page 9

where the reflection coefficient [Fig. 3 (a)] of the effective medium vanishes. In addition to

these modes, different Brewster modes appear depending on the value of filling factor. Also,

we see on the reflection curves (Fig. 3) that the Christiansen point [35] for which the reflectivity

is zero for all κ does not depend on the porosity. Indeed, an inspection of expressions (13) and

(14) shows that the condition for the Christiansen point of the host material εh= 1 implies that

ε?= 1 and ε⊥= 1 so that, according to (17), the reflection coefficients vanish.

5.Heat flux between porous media

Before we discuss the influence of the inclusions on the heat flux, we show in Fig. 4 the results

of the mean Poynting vector ?Sz? between two semi-infinite SiC bodies at fixed temperatures

T1= 300K and T2= 0K. First of all one can see that the heat flux becomes very large for

distances much smaller than the thermal wavelength λth= ¯ hc/kBT (which is about 7.68µm

for T = 300K). At d = 10nm the heat flux for the two SiC bodies is about 1000 times larger

than the heat flux between two black bodies. This increase is due to the frustrated total internal

reflection and to the coupled surface phonon polariton modes [9]. In the propagating regime,

i.e., for distances larger than λththe heat flux is determined by Kirchhoff-Planck’s law and is

limited by the black-body value. Note, that the heat flux is dominated by the p-polarized modes

for distances smaller than 100nm and larger than 10µm, whereas for distances in between it is

dominated by the s-polarized modes.

0.1

1

10

100

1000

10-8

10-7

10-6

10-5

10-4

< Sz > / SBB

d (meters)

s + p

s

p

Figure 4. Heat flux between two SiC plates over distance with T1=300K and T2=0K. The

flux is normalized to the value for two black bodies SBB= 459.6Wm−2. The contribution

of the s- and p-polarized part are shown as well.

Now, we introduce the inclusions by using the Maxwell-Garnett expression in Eq. (13) and

(14). We use the same filling factor for both materials, so that we have a symmetric situation.

In Fig. 5 we show the resulting heat flux normalized to the values for the two non-pourous SiC

plates shown in Fig. 4. We find that for distances smaller than 100nm and larger than 1µm the

heat flux becomes larger when we add air inclusions, whereas for intermediate distances the

heat flux is reduced.

In order to see how the s- and p-mode contribution is changed by the porosity, we show in

Fig. 6 (a) and (b) the plots for the separate contributions of s- and p-polarized modes. It is clear

that the p-polarized part of the heat flux gets enhanced for all distances when compared to the

isotropic case, regardless of the filling factor. The s-polarized part in turn gives a larger heat flux

for distances larger than about 1µm and a smaller heat flux for distances smaller than 1µm.

Therefore, the smaller heat flux found in Fig. 5 for intermediate distances is associated to the

dominance of s-polarized modes in that distance regime.

Page 10

0.75

1

1.25

1.5

10-8

10-7

10-6

10-5

10-4

< Sz > / < Sz,iso >

d (meters)

f = 0.1

f = 0.3

f = 0.5

Figure 5. Heat flux between two porous SiC plates over distance with T1= 300K and

T2= 0K. The flux is normalized to the value for two SiC plates shown in Fig. 4.

In summary, by introducing inclusions we find for large and small distances an increase

of the heat flux. For the propagating regime (d > λth) this can be understood from a simple

argument: the vacuum holes simply dilute the material so that, according to Kirchhoff’s law,

the reflectivity is decreased and hence the emissivity is increased. In fact, for f = 1 one would

retrieve the black body result, since in this case the reflectivity is zero. On the other hand,

there is no such simple argument for the increased heat flux in the near-field region. Here, it

is necessary to study how the coupled surface modes, which give the main contribution to the

heat flux for distances smaller than 100nm, are influenced by the introduction of the inclusions.

This will be done by inspection of the transmission coefficient in the next section.

6.Transmission coefficient

As mentioned before, for the small distance regime (d < 100nm) the heat flux between two

isotropic semi-infinite SiC-bodies is solely dominated by the p-polarized contribution. This

remains true for the porous SiC bodies. In fact, the dominance of the p-polarized contribution

becomes even greater with increasing filling factors. Hence, to understand the observation that

by introducing some porosity the heat flux becomes larger, it suffices to study the p-polarized

contribution.

In Fig. 7 we show the transmission coefficient Tp(ω,κ;d) in the (ω,κ)-plane for different

filling factors and a distance d = 100nm. In Fig. 7 (a) one can see Tp(ω,κ;d) for two isotropic

SiC plates. Here, Tp(ω,κ;d) is one or close to one for the propagating modes, the total inter-

nal reflection modes and the coupled surface phonon polaritons. In the plotted region one can

mainly see the coupled surface phonon polaritons, which are responsible for the large heat flux

at small distances. Now, for f =0.1 one can see in Fig. 7 (b) that a second coupled surface mode

appears due tothe airinclusions. Inaddition, thecoupled surface mode of the bulkSiC isshifted

to smaller frequencies. When increasing the filling factor [Fig. 7 (c) and (d)] the upper coupled

surface modes shift to higher frequencies and become less important for the transmission coef-

ficient. On the other hand, the low frequency surface modes shift further to lower frequencies.

Between the two coupled surface mode branches a band of frustrated internal reflection modes

is formed which gives also a non-negligible contribution to the transmission coefficient.

In order to get further information we now consider the spectral mean Poynting vector ?Sω?

defined in Eq. (7) for p-polarization only. We have plotted this quantity in Fig. 8 at the same

distance as before, i.e., d = 100nm, and again for different filling factors. As in Fig. 7 one can

see the strong contribution of the two coupled surface mode resonances, which are shifted in

Page 11

1

1.25

1.5

10-8

10-7

10-6

10-5

10-4

< Sz > / < Sz,iso >

d (meters)

f = 0.1

f = 0.3

f = 0.5

0.25

0.5

0.75

1

1.25

1.5

10-8

10-7

10-6

10-5

10-4

< Sz > / < Sz,iso >

d (meters)

f = 0.1

f = 0.3

f = 0.5

(b)

(a)

Figure 6. As in Fig 5 but for the (a) s- and (b) p-polarized contribution only.

frequencieswhenchangingthefillingfactor.Moreover,theshiftingoftheprimarysurfacemode

to lower frequencies by itself also enhances the flux, as such a shift brings that surface mode

closer to the peak wavelength of blackbody radiation as given by the Wien’s law. Furthermore,

one can now observe, that when increasing the filling factor the low frequency resonance is not

only shifted to smaller frequencies, but the resonance is also getting stronger.

The study can now be completed when considering the mean transmission factor for the

p-polarized modes, that was introduced in Ref. [9] as

Tp(κ) =

3

π2

?∞

0

du f(u)Tp(u,κ;d)

(24)

with u = ¯ hω/kBT and f = u2eu/(eu−1)2. It represents the mean transmission coefficient of

a mode specified by it’s wave vector κ for a given temperature T and a small temperature

difference ∆T between the two bodies. By means of this quantity the heat flux can be rewritten

in a Landauer-like form [9]

?Sz? =π2

3

k2

BT

h

?dκ

2πκTp(κ)∆T.

(25)

Note, that for κd ≫ 1 and κ > ω/c the transmission coefficient Tp(ω,κ;d) is exponentially

damped [see Eq. (8)] and therefore also the mean transmission factor Tp(κ). This damping

determines the wave vector cutoff and hence the number of states contributing to the heat flux.

Now, in Fig. 9 we plot Tp(κ) for a given distance of d = 100nm and different filling factors

normalized to the mean transmission factor for two semi-infinite SiC bodies. For f = 0.1 the

Page 12

(a)

(b)

(c)

(d)

Figure 7. Transmission coefficient Tp(ω,κ;d) in the (ω,κ)-plane for two porous SiC slabs

with different filling factors (a) f =0, (b) f =0.1, (c) f =0.3, and (d) f =0.5. The distance

is fixed at d = 100nm.

0

1.65

2

4

6

8

1.71.75

ω / 1014 s-1

1.81.851.9

< Sω > / 1013 m-2

0

0.1

0.3

0.5

Figure 8. Spectral mean Poynting vector ?Sω? defined in Eq. (7) for two porous SiC slabs

with different filling factors f = 0,0.1,0.3,0.5 considering only the p-polarized contribu-

tion. The distance is fixed at d = 100nm.

mean transmission coefficient for the porous SiC increases for intermediate κ but decreases for

very large κ. The increased mean transmission factor is due to the second coupled surface mode

and the frustrated modes, whereas the lower value for large wave vectors can be attributed to

a stronger cutoff in the transmission coefficient, which means that the number of contributing

modes is decreased. The enhancement of the transmission factor due to the surface mode pre-

vails and leads to an enhanced heat flux at that distance. The same mechanism is responsible for

the enhanced heat flux for f =0.3. On the other hand, for larger filling factors the curves change

slightly for intermediate κ compared to the curve for f = 0.3. The contribution in that interme-

Page 13

0.5

1

1.5

2

2.5

0.01 0.1 1 10

T / Tiso

κ d

0.1

0.2

0.3

0.4

0.5

Figure 9. Mean transmission coefficient defined in Eq. (24) for different filling factors nor-

malized to the isotropic case (f = 0). The distance is fixed at d = 100nm and the tempera-

ture at T = 300K.

diate region is due to the second coupled surface mode branch and the frustrated modes. But for

very large κ the mean transmission coefficient increases compared to f = 0.3. This means that

by introducing a higher porosity we soften the cutoff of the transmission coefficient. Hence, the

number of modes contributing to the heat flux is increased and results for large filling factors in

a further enhanced heat flux.

The dependence of the cutoff on the filling factor for large κ can easily be discussed for the

transmission coefficient Tp(κ,ω;d). It was found in Ref. [9] that the cutoff region, i.e., where

Tp(κ,ω;d) is exponentially damped, is given by

κiso> log

?

2

Im(ε)

?1

2d

(26)

when considering two isotropic semi-infinite bodies at the surface mode resonance frequency

[see also Refs. [36]]. For the uniaxial anisotropic case as considered here, this relation changes

to

κuni> log

Im(√ε?ε⊥)

where the permittivities have to be evaluated at the surface mode resonance frequency of the

semi-infinite anisotropic body (see Appendix B). In Fig. 10 we show a plot of κuni/κisoover

the filling factor. It is seen that by introducing the air inclusions the cutoff first decreases and

then monotonically increases. This is the same qualitative behavior as observed for the mean

transmission factor Tp(κ) in Fig. 9 for κd ≫ 1. This reasoning confirms that the number of

contributing modes is the main mechanism for increasing the heat flux at small distances and

large filling factors (f > 0.3).

?

2

?1

2d

(27)

7. Conclusion

We have presented a detailed study of near and far field heat transfer between two flat uni-

axial media made of polar materials (in our case, SiC) in which cylindrical inclusions drilled

orthogonally to surfaces are uniformally distributed.

After applying the classical stochastic electrodynamic theory to anisotropic materials we

haveshownthat,forshortdistances,theheatfluxbetweensuchmediacanbesignificantlylarger

than those traditionally measured between two isotropic materials in the same non-equilibrium

Page 14

0.84

0.88

0.92

0.96

1

1.04

0 0.1 0.2

filling factor f

0.3 0.4 0.5

κuni / κiso

Figure 10. Plot of the normalized cutoff value κuni/κisoover filling factor f.

thermal conditions. For small filling factors we have determined that this enhancement stems

from additional surface waves arising at the uniaxial material-vacuum interface, clearly indi-

cating that such increase is intrinsically connected to anisotropy. Indeed, we did calculations

for isotropically rarified SiC plates with low filling factors (f ≤ 0.1) and found that the heat

transfer modification for is much smaller. In contrast, for larger filling factors (f > 0.3) we

have shown that, after a thorough analysis of the transmission factor, the enhancement in heat

transfer arises mainly from the increased number of modes contributing to the flux.

We thank Henri Benisty for very helpful discussions on the subject of homogenization. S.-A.

B.gratefullyacknowledges supportfromtheDeutscheAkademiederNaturforscherLeopoldina

(Grant No. LPDS 2009-7). This research was partially supported by Triangle de la Physique,

under the contract 2010-037T-EIEM.

A.Green’s dyadic in the gap region

In order to construct the Green’s dyadic in the vacuum gap we first start with the Green’s dyadic

in free space. If z > z′Weyl’s expansion for the Green’s dyadic is [37]

G(r,r′) =

?

d2κ

(2π)2

ieiκ·(x−x′)

2γr

eiγr(z−z′)1

(28)

with γr=?ω2/c2−κ2, x = (x,y)tand κ = (kx,ky)t. The unit dyadic 1 is the unit dyadic in the

1 = ˆ a+

polarization basis and is defined as

s⊗ ˆ a+

s+ ˆ a+

p⊗ ˆ a+

p.

(29)

The polarisation vectors for s- and p-polarized waves are given by

ˆ a+

s=1

κ

−ky

kx

0

and

ˆ a+

p=

c

κω

kxγr

kyγr

−κ2

.

(30)

By construction both polarization vectors are orthogonal. For propagating waves they are also

normalized. The Fourier component G(κ;z,z′) of the Greens dyadic is defined by

G(r,r′) =

?

d2κ

(2π)2G(κ;z,z′)eiκ·(x−x′).

(31)

Page 15

The above expression for the Green’s dyadic represents the field of a right going wave at z

of a source of unit strength placed at z′. If a semi-infinite medium is located at z > d then this

wave will be reflected so that the Green’s dyadic G(κ;z,z′) reads at z > z′

GA(κ;z,z′) =

i

2γr

?1eiγr(z−z′)+e2iγrdeiγr(z+z′)R2

?

(32)

where we have introduced the reflection matrix

R2= ∑

i,j={s,p}

r2

i,jˆ a−

i⊗ ˆ a+

j

(33)

with the reflection coefficients r2

and ˆ a−

defined as

i,jand the polarization vectors ˆ a−

p= −c/(κω)(kxγr,kyγr,κ2)t

s= ˆ a+

s. If there is now a second semi-infinite medium at z < 0 with a reflection operator

R1= ∑

i,j={s,p}

r1

i,jˆ a+

i⊗ ˆ a−

j

(34)

the waves in that cavity will be multiply reflected at the boundaries at z = 0 and z = d so that

[23]

GA(κ;z,z′) =

i

2γr

?

1eiγr(z−z′)+e2iγrde−iγr(z+z′)R2

+e2iγrdeiγr(z−z′)R1R2

+e4iγrde−iγr(z+z′)R2R1R2+...

?

.

(35)

Summing up all contributions we get

GA(κ;z,z′) =

i

2γr

?

D12eiγr(z−z′)+D21R2e2iγrde−iγr(z+z′)

?

(36)

where we have introduced

D12= (1−R1R2e2iγrd)−1,

D21= (1−R2R1e2iγrd)−1.

(37)

(38)

The expression in Eq. (36) is not yet the complete intracavity Green’s dyadic, since we have

not considered the waves which start from z′as left going waves and arrive after being reflected

at the boundary at z = 0 at z > z′. With the same reasoning as for GA(κ;z,z′) we find for this

contribution

GB(κ;z,z′) =

i

2γr

?

D12R1eiγr(z+z′)+D21R2R1e2iγrdeiγr(z′−z)

?

(39)

Finally, the intracavity Green’s dyadic is given by the sum of Eq. (36) and (39) yielding

Gintra=

i

2γr

?

D12

?

1eiγr(z−z′)+R1eiγr(z+z′)

?

+D21

?

R2R1eiγr(z′−z)e2iγrd+R2e2iγrde−iγr(z+z′)

??

(40)